Optimization based controller tuning systems and methods

ABSTRACT

One embodiment of the present disclosure describes an industrial system, which includes a control system that controls operation of an industrial process by instructing an automation component in the industrial system to implement a manipulated variable setpoint. The control system includes a process model that model operation of the industrial process, control optimization that determines the manipulated variable setpoint based at least in part on the process model, a control objective function, and constraints on the industrial process, in which the control objective function includes a tuning parameter that describes weighting between aspects of the industrial process affected by the manipulated variable setpoint; and tuning optimization circuitry that determines the tuning parameter based at least in part on a tuning objective function, in which the tuning objective function is determined based at least in part on a closed form solution to an augmented version of the control objective function, which includes the constraints as soft constraints.

BACKGROUND

The present disclosure relates generally to control systems and, moreparticularly, to tuning of control systems.

Generally, a control system may facilitate controlling operation of aprocess, for example, in an industrial plant or an industrial automationsystem. More specifically, the control system may determine manipulatedvariable setpoints, which when implemented control operation of theprocess. For example, a model predictive control system may determinemanipulated variable setpoints based at least in part on to transitionthe process from a current operating state to a desired operating stateover a control horizon (e.g., future time steps).

In some embodiments, the control system may determine the manipulatedvariable setpoints based on tuning parameters of the control system. Forexample, in a model predictive control system, the tuning parameters maydescribe weighting between deviation of a controlled variable from adesired value, deviation of a manipulated variable from a desired value,and rate of change of the manipulated variable. In other words,transitioning the process to the desired operating state may befacilitated by proper tuning of the tuning parameters.

BRIEF DESCRIPTION

Certain embodiments commensurate in scope with the originally claimedembodiments are summarized below. These embodiments are not intended tolimit the scope of the claimed invention, but rather these embodimentsare intended only to provide a brief summary of possible forms of thesystems and techniques described herein. Indeed, the systems andtechniques described herein may encompass a variety of forms that may besimilar to or different from the embodiments set forth below.

One embodiment of the present disclosure describes an industrial system,which includes a control system that controls operation of an industrialprocess by instructing an automation component in the industrial systemto implement a manipulated variable setpoint. The control systemincludes a process model that model operation of the industrial process;control optimization that determines the manipulated variable setpointbased at least in part on the process model, a control objectivefunction, and constraints on the industrial process, in which thecontrol objective function includes a tuning parameter that describesweighting between aspects of the industrial process affected by themanipulated variable setpoint; and tuning optimization circuitry thatdetermines the tuning parameter based at least in part on a tuningobjective function, in which the tuning objective function is determinedbased at least in part on a closed form solution to an augmented versionof the control objective function, which includes the constraints assoft constraints.

Another embodiment of the present disclosure describes a method forcontrolling operation of an industrial process that includesdetermining, using a control system, soft constraints based at least inpart on constraints on manipulated variables, controlled variables, orboth of the industrial process; determining, using the control system,an augmented objective function based at least in part on a controlobjective function and the soft constraints; determining, using thecontrol system, a closed form solution to the augmented objectivefunction, in which the closed form solution is a function of tuningparameters in the control objective function; determining, using thecontrol system, a tuning objective function based at least in part onthe closed form solution and the control objective function;determining, using the control system, a first set of the tuningparameters that minimize the tuning objective function; determining,using the control system, manipulated variable setpoints based at leastin part on the control objective function, in which the controlobjective function comprises the tuning parameters as weighting onaspects of the industrial process affected by manipulated variables ofthe industrial process, controlled variables of the process, or both;and controlling, using the control system, operation of the industrialprocess by instructing one or more automation components to implementthe manipulated variable setpoints.

Another embodiment of the present disclosure describes A tangible,non-transitory, computer-readable medium that stores instructionsexecutable by a processor of a control system. The instructions includeinstructions to instruct, using the processor, state transitionoptimization circuitry to determine a desired operating trajectory of aprocess over a control horizon that transitions the process from acurrent operating state to a desired operating state after the controlhorizon; instruct, using the processor, control optimization circuitryto determine an actual operating trajectory of the process to implementin the process that minimize a control objective function subject tofirst constraints on the process, in which the control objectivefunction comprises a first tuning parameter that weights cost associatedwith value of the actual operating trajectory and a second tuningparameter that weights cost associated with changes in the actualoperating trajectory at each of a plurality time steps in the controlhorizon; and instruct, using the processor, tuning optimizationcircuitry to determine at least the first tuning parameter and thesecond tuning parameter to minimize a tuning objective function, inwhich the tuning objective function is determined based at least in parton a closed form solution to an augmented version of the controlobjective function with the first constraints included as softconstraints.

DRAWINGS

These and other features, aspects, and advantages of the presentdisclosure will become better understood when the following detaileddescription is read with reference to the accompanying drawings in whichlike characters represent like parts throughout the drawings, wherein:

FIG. 1 is a schematic diagram of a process and a control system, inaccordance with an embodiment of the present disclosure;

FIG. 2 is a block diagram of the control system of FIG. 1 with a processmodel, state transition optimization circuitry, control optimizationcircuitry, and tuning optimization circuitry, in accordance with anembodiment of the present disclosure;

FIG. 3 is a block diagram of the process model of FIG. 2, in accordancewith an embodiment of the present disclosure;

FIG. 4 is a block diagram of the tuning optimization circuitry of FIG.2, in accordance with an embodiment of the present disclosure;

FIG. 5 is a block diagram of the control optimization circuitry of FIG.2, in accordance with an embodiment of the present disclosure;

FIG. 6 is a flow diagram describing a process for controlling operationof the process of FIG. 1 by using the control optimization circuitry ofFIG. 5, in accordance with an embodiment of the present disclosure;

FIG. 7 is a block diagram of the tuning optimization circuitry of FIG.2, in accordance with an embodiment of the present disclosure; and

FIG. 8 is a flow diagram describing a process for determining tuningparameters using the tuning optimization circuitry of FIG. 7, inaccordance with an embodiment of the present disclosure.

DETAILED DESCRIPTION

One or more specific embodiments of the present disclosure will bedescribed below. In an effort to provide a concise description of theseembodiments, all features of an actual implementation may not bedescribed in the specification. It should be appreciated that in thedevelopment of any such actual implementation, as in any engineering ordesign project, numerous implementation-specific decisions must be madeto achieve the developers' specific goals, such as compliance withsystem-related and business-related constraints, which may vary from oneimplementation to another. Moreover, it should be appreciated that sucha development effort might be complex and time consuming, but wouldnevertheless be a routine undertaking of design, fabrication, andmanufacture for those of ordinary skill having the benefit of thisdisclosure.

When introducing elements of various embodiments of the presentdisclosure, the articles “a,” “an,” “the,” and “said” are intended tomean that there are one or more of the elements. The terms “comprising,”“including,” and “having” are intended to be inclusive and mean thatthere may be additional elements other than the listed elements.

Control systems are often used to control operation of a process, forexample, in automation systems, automation plants, factories, and thelike. More specifically, the control system may control operation of theprocess by instructing automation components to implement an operatingtrajectory (e.g., manipulated variable setpoints) to achieve a desiredoperating state. For example, in a manufacturing process, the controlsystem may instruct a conveyer belt motor (e.g., an automationcomponent) to actuate at a particular speed (e.g., a manipulatedvariable setpoint) to achieve a desired output production (e.g., adesired operating state). In other words, the control system maydetermine the operating trajectory (e.g., manipulated variablesetpoints) such that the process transitions from a current operatingstate (e.g., current manipulated variables and controlled variables) tothe desired operating state.

In some embodiments, to facilitate determining the operating trajectory,the control system may include tuning parameters. More specifically, thetuning parameters may govern weighting between various aspects of theprocess affected by the operating trajectory. For example, in a modelpredictive control system, the control system may include an objectivefunction with tuning parameters. In some embodiments, the tuningparameters in the objective function may weight deviation of acontrolled variable from a desired value, deviation of a manipulatedvariable form a desired value, and change in the manipulated variable.In other words, proper tuning of the tuning parameters facilitatesdetermination of the operating trajectory that enable realizing adesired operating state of the process.

It may be possible to determine the tuning parameters heuristically. Forexample, a training sequence may transition the process from a currentoperating state to a desired operating state and determine manipulatedvariable setpoints to achieve the transition. By running the processthrough multiple iterations of the training sequence, tuning parametersmay be heuristically determined to capture the relationship between thecurrent operating state, the desired operating state, and themanipulated variable setpoints to achieve the each transition.

However, since the process is run through multiple iterations of thetraining sequence, heuristically determining the tuning parameters isgenerally an offline task. In other words, the tuning parameters may bepredetermined before deploying the control system to control theprocess. Additionally, since each process may vary, the relationshipused to tune the tuning parameters is generally determined by a controlengineer with sufficient knowledge of the process and the controlsystem. In other words, the determination of the tuning parameters maybe based largely on the knowledge and skill of a control engineer. Assuch, heuristically determining the tuning parameters may generally belimited to offline determination and by the skill/knowledge of a controlengineer.

Accordingly, as will be described in more detail below, the techniquesdescribed in the present disclosure may improve tuning of the tuningparameters by enabling the tuning parameters to be determined online ina systematic manner. In some embodiments, a control system may includestate transition optimization circuitry, control optimization circuitry,and tuning optimization circuitry. More specifically, the control systemmay use the state transition optimization circuitry to determine adesired trajectory of controlled variables and manipulated variables ofa process over a control horizon.

Based at least in part on the desired trajectories, the control systemmay use the control optimization circuitry to determine manipulatedvariable setpoints to implement in the process. In some embodiments, thecontrol optimization circuitry may determine the manipulated variablesetpoints based at least in part on a control objective function. Forexample, the control optimization circuitry may determine manipulatedvariable setpoints that minimize the control objective function subjectto constraints on the process. In some embodiments, the controlobjective function may include tuning parameters that describe weightingof factors in the objective function, such as a cost associated withdeviation of a controlled variable from the desired trajectory, a costassociated with deviation of a manipulated variable from the desiredtrajectory, and a cost associated with changes to the manipulatedvariable.

To determine the tuning parameters, the control system may use thetuning optimization circuitry. In some embodiments, the tuningoptimization circuitry may determine a closed form solution to anaugmented objective function, which may be determined based at least inpart on the control objective function. The closed form solution to theaugmented objective function may parameterize the manipulated variablesas a function of the tuning parameters in the augmented objectivefunction. More specifically, the augmented objective function may be anaugmented version of the control objective function with the constraintsincluded as soft constraints. For example, the soft constraints may beincluded by introducing slack variables and/or transforming theconstraints into barrier functions. In fact, including the softconstraints may enable further weighting between various constraints. Inother embodiments, the control system may determine the closed formsolution of the control objective function while disregarding theconstraints on the process.

Based on the closed form solution, the control system may determine atuning objective function. For example, in some embodiments, the controlsystem may determine the tuning objective function by substituting theclosed form solution into the control objective function so that thetuning objective function is a function of the tuning parameters. Thus,by solving the tuning objective function, the control system maydetermine the tuning parameters. For example, in some embodiments, thecontrol system may determine the tuning parameters based on minimizationof the tuning objective function. As discussed above, the control systemmay then utilize the tuning parameters to determine manipulated variablesetpoints to implement in one or more automation components, therebycontrolling operation of the process.

Accordingly, since the tuning parameters may be determined based merelyon the control objective function, the current operating state of theprocess, the desired operating state of the process, the process model,and any constraints on the process, the techniques described in thepresent disclosure provide a systematic approach to determining thetuning parameters. In fact, the techniques described herein may beapplicable to a variety of processes, process models, and controlsystems combinations, for example, without reliance on heuristictraining sequences and control engineer skill/knowledge. Furthermore,the techniques described herein may enable the tuning parameters to bedetermined in an online manner (e.g., after deployment of the controlsystem to control the process). Moreover, the techniques describedprovide for the use of parametric hybrid models to enable systematictuning of an optimization-based control system (e.g. model predictivecontrol system described in FIG. 2) for use with (e.g., control of)nonlinear processes. For example, the parametric hybrid model may enablethe optimization-based control system to determine a closed-formsolution (e.g., in process block 78 of FIG. 7) indicating transition ofoperating condition when a controlled process is a nonlinear process.

To help illustrate, one embodiment of a process system 10 is describedin FIG. 1. As depicted, the process system 10 includes a process 12 anda control system 14. In some embodiments, the process 12 may be anyconceivable type, such as a manufacturing process, a steady stateprocess, a batch process, a chemical process, a material handlingprocess, an energy utilizing process, an energy production process, orany combination thereof.

Thus, as depicted the process 12 may receive one or more inputs 16 usedto produce one or more outputs 18. For example, the inputs 16 mayinclude feed stock, electrical energy, fuel, parts, assemblies,sub-assemblies, or any combination thereof. Additionally, the outputs 18may include finished products, semi-finished products, assemblies,manufacturing products, by products, product properties, or anycombination thereof.

To produce the one or more outputs 18, the control system 14 may controloperation of the process 12. More specifically, the control system 14may control operation by outputting control signals to instruct one ormore automation components 20 to implement manipulated variablesetpoints. In some embodiments, the automation components 20 may includecontrollers, input/output (I/O) modules, motor control centers, motors,human machine interfaces (HMIs), operator interfaces, contactors,starters, drives, relays, protection devices, switchgear, compressors,scanners, gauges, valves, flow meters, and the like. For example, thecontrol system 14 may instruct a motor (e.g., an automation component20) to actuate at a particular speed (e.g., a manipulated variablesetpoint).

In some embodiments, the control system 14 may determine the manipulatedvariable setpoints based at least in part on operational parameterdetermined by one or more sensors 22. More specifically, the sensors 22may communicate measurement signals informing the control system 14 ofthe determined operational parameters. In some embodiments, theoperational parameters may include temperature, flow rate, electricalpower, and the like. For example, a temperature sensor 22 may inform thecontrol system 14 regarding temperature (e.g., an operational parameter)of a motor (e.g., an automation component 20). In fact, in someembodiments, the operational parameters may include information enablingthe control system 14 to determine a current operating state of theprocess 12 (e.g., current manipulated variables and/or controlledvariables).

Accordingly, the control system 14 may include a processor 24 and memory26 to facilitate performing operations, such as determining manipulatedvariable setpoints and/or determining current operating state of theprocess 12. More specifically, the processor 24 may execute instructionand/or process data stored in memory 26. As such, the processor 24 mayinclude one or more general purpose microprocessors, one or moreapplication specific processors (ASICs), one or more field programmablelogic arrays (FPGAs), or any combination thereof. Additionally, thememory 26 may include random access memory (RAM), read only memory(ROM), rewritable flash memory, hard drives, optical discs, and thelike.

Furthermore, the control system 14 may include controllers to facilitateenabling different types of control schemes. For example, the controlsystem 14 may include one or more model predictive control (MPC)controllers, one or more proportional-integral-derivative (PID)controllers, one or more neural network controllers, one or more fuzzylogic controllers, or any combination thereof. Generally, in each of thevarious control schemes, the control system 14 may determine manipulatedvariable setpoints based at least in part on tuning parameters. Forexample, in a PID control system, the control system 14 may utilizetuning parameters, such as a proportional gain, an integral gain, and aderivative gain. Additionally, in a MPC control system, the controlsystem 14 may utilize tuning parameters in an objective function toweight aspects of the process 12. For example, the tuning parameters mayweight deviation of a controlled variable from a desired value,deviation of a manipulated variable form a desired value, and change inthe manipulated variable.

One of ordinary skill in the art should recognize that the techniquesdescribed herein may facilitate determining tuning parameters for acontrol system 14 utilizing any suitable control scheme. However, tosimplify discussion, the techniques described herein will be describedin regard to a model predictive control system 14. Generally, a modelpredictive control system 14 may determine manipulated variablesetpoints accounting for operation of the process 12 over a controlhorizon (e.g., manipulated variable trajectories and/or controlledvariable trajectories over future time steps).

To help illustrate, one embodiment of a model predictive control system14A is described in FIG. 2. As depicted, the model predictive controlsystem 14A includes a process model 28, state transition optimizationcircuitry 30, control optimization circuitry 32, and tuning optimizationcircuitry 34. More specifically, the process model 28 may modeloperation of the process 12, for example, by describing a relationshipbetween manipulated variables and resulting controlled variables. Thus,as will be described in more detail below, the process model 28 may beused to facilitate operation of the state transition optimizationcircuitry 30, the control optimization circuitry 32, the tuningoptimization circuitry 34, or any combination thereof. Additionally,since the techniques described herein may be implemented online (e.g.,after deployment of the control system 14 to control the process 12), itmay be beneficial to utilize a computationally efficient process model28. One example of such a process model 28 is a parametric hybrid model.

To help illustrate, one embodiment of a parametric hybrid model 28A isdescribed in FIG. 3. As depicted, the parametric hybrid model 28Aincludes an empirical model 36, a parameter model 38, and a parametricfirst-principles model 40. Additionally, the parametric hybrid model 28Areceives manipulated variables u_(k) from the control system 14. Asdiscussed above, the manipulated variables u_(k) may be determined bythe control system 14 based at least in part on measurements by thesensors 22. Additionally or alternatively, the manipulated variablesu_(k) may be determined by the control system 14 based at least in parton a determined trajectory of the manipulated variables over a controlhorizon.

Based on the manipulated variables u_(k), the empirical model 36 maydetermine empirical model outputs w_(k). For example, in the depictedembodiment, the empirical model 36 may determine the empirical modeloutputs w_(k) as a function of the manipulated variables u_(k) andempirical model parameters p. Additionally, based on the empirical modeloutputs w_(k) and the manipulated variables u_(k), the parameter model38 may determine fundamental model parameters θ_(k). For example, in thedepicted embodiment, the parameter model 38 may determine thefundamental model parameters θ_(k) as a function of the manipulatedvariables u_(k) and the empirical model outputs w_(k). In fact, in someembodiments, the fundamental model parameters θ_(k) may be identical tothe empirical model outputs w_(k) in their simplest form.

Based on the fundamental model parameters θ_(k) and the manipulatedvariables u_(k), the parametric first-principles model 40 may determinestate variables x_(k) and controlled variables y_(k). For example, inthe depicted embodiment, the parametric first-principles model 40 maydetermine the state variables x_(k) as a function of the manipulatedvariables u_(k), a previous state variable x_(k−1), and the empiricalmodel output w_(k). In some embodiments, the state variables x_(k) mayinclude any combination of measured and/or unmeasured operationalparameters (e.g., manipulated variables and/or controlled variables) ofthe process 12.

Additionally, in the depicted embodiment, the parametricfirst-principles model 40 may determine the controlled variables y_(k)as a function of the manipulated variables u_(k), the current statevariable x_(k), and the empirical model output w_(k). As such, theparametric hybrid model 28A may be defined as follows:w _(k) =f ₁(u _(k) ,p);  (1)θ_(k) =f ₂(u _(k) ,w _(k));  (2)x _(k) =F(u _(k) ,x _(k−1),θ_(k)); and  (3)y _(k) =G(u _(k) ,x _(k),θ_(k));  (4)where u_(k) is a vector of the manipulated variables at time step k, pis a vector of empirical model parameters, w_(k) is a vector ofempirical model outputs at time step k, θ_(k) is a vector of fundamentalmodel parameters at time step k, x_(k) is a vector of state variables attime step k, x_(k−1) is a vector of state variables at time step k−1,and y_(k) is a vector of controlled variables at time step k. In thismanner, the parametric hybrid model 28A may enable the model predictivecontrol system 14A to predict the controlled variables at a time stepbased at least in part on the manipulated variables at the time step andthe state variable at a previous time step (e.g., manipulated variablesand/or controlled variables at the previous time step).

As described above, the process model 28 may be utilized to facilitateexecuting explicit real-time optimization (e.g., state transitionoptimization circuitry 30, control optimization circuitry 32, and/ortuning optimization circuitry 34). As such, the process model 28 shouldbe sufficiently efficient to enable real-time optimization. In someembodiments, a parametric hybrid model 28A may be particularly suitableby providing a systematic approach to balancing model accuracy andcomputational efficiency (e.g., execution speed).

For example, in some embodiments, the parametric hybrid model 28A mayvary level of detail depending on desired computational complexity. Tohelp illustrate, the parametric hybrid model 28A may include acombination of steady-state models, which may be a linear representationof the process 12 at steady-state, and dynamic models, which may be anon-linear representation of the process 12 during transitions. As such,the steady-state model type may less accurately represent the process12, but be less computationally complex, while the dynamic model typemay more accurately represent the process 12, but be morecomputationally complex. As such, the parametric hybrid model 28A maydetermine the model type (e.g., for the steady-state models or thedynamic models) to utilize in one or more of the empirical model 36, theparameter model 38, and the parametric first-principles model 40. Inthis manner, the parametric hybrid model 28A provides a systematicframework for trading off model accuracy and computational efficiency,thereby enabling real-time optimization (e.g., state transitionoptimization circuitry 30, control optimization circuitry 32, and/ortuning optimization circuitry 34).

As discussed above, the state optimization circuitry 30 may use theprocess model 28. More specifically, the state transition optimizationcircuitry 30 may determine a desired operating trajectory of the process12 to transition the process 12 from a current operating state to adesired operating state over a control horizon. In some embodiments, thedesired operating trajectory may include a desired trajectory of themanipulated variables and a desired trajectory of the controlledvariables over the control horizon.

To help illustrate, one embodiment of the state transitions optimization30 is described in FIG. 4. As depicted, the state transitionoptimization circuitry 30 receives a desired operating state 42 and acurrent operating state, which includes current manipulated variables 44and controlled variables 46. Additionally, the state transitionoptimization circuitry 30 outputs the desired controlled variabletrajectories 48 and the desired manipulated variable trajectories 50.More specifically, the desired controlled variable trajectories 48 andthe desired manipulated variable trajectories 50 may describe desiredoperation of the process 12 over the control horizon to transition fromthe current operating state to the desired operating state.

To facilitate determining the desired controlled variable trajectories48 and the desired manipulated variable trajectories 50, the modelpredictive control system 14A may utilize the process model 28, whichmay be parametric hybrid model 28A or the like. As discussed above, theprocess model 28 may describe expected operation of the process 12.Thus, the process model 28 may enable the model predictive controlsystem 14A to determine various sets of manipulated variable andcontrolled variable trajectories that are expected to transition theprocess 14 from the current operating state to the desired operatingstate 42.

Based at least in part on a transition objective function 52, the modelpredictive control system 14A may identify the desired control variabletrajectories 48 and the desired manipulated variable trajectories 50from the sets trajectories. More specifically, the transition objectivefunction 52 may describe costs associated with the value of themanipulated variables, costs associated with value of the controlledvariables, costs associated with changes to the manipulated variables,and/or costs associated with changes to the controlled variables at eachtime step of the control horizon. As such, using the transitionobjective function 52, the model predictive control system 14A maydetermine a cost associated with each set of trajectories and identifythe set with the lowest associated cost as the desired control variabletrajectories 48 and the desired manipulated variable trajectories 50.

In other embodiments, the desired control variable trajectories 48 andthe desired manipulated variable trajectories 50 may be supplieddirectly to the model predictive control system 14A, for example by auser. In such embodiments, the state transition optimization circuitry30 may be obviated.

Generally, constraints on the process 12 may limit the ability of theprocess 12 to implement exactly the desired operating trajectory (e.g.,control variable trajectories 48 and the desired manipulated variabletrajectories 50). In some embodiments, the constraints may limit valueof one or more manipulated variables, value of one or more controlledvariables, rate of change of one or more manipulated variables, and/orrate or change of one or more controlled variables. Accordingly, themodel predictive control system 14A may use the control optimizationcircuitry 32 to determine an actual operating trajectory (e.g.,manipulated variable setpoints) subject to the constraints. For example,in some embodiments, the control optimization circuitry 32 may determinemanipulated variable setpoints to be implemented at each time step overthe control horizon (e.g., manipulated variable setpoint trajectories)subject to constraints on the process 12.

To help illustrate, one embodiment of the control optimization circuitry32 is described in FIG. 5. As depicted, the control optimizationcircuitry 32 receives the desired operating trajectory, which includesthe desired control variable trajectories 48 and the desired manipulatedvariable trajectories 50 over a control horizon, and the currentoperating state, which includes the manipulated variables 44 andcontrolled variables 46 determined for a current time step.Additionally, the control optimization circuitry 32 outputs themanipulated variable setpoint trajectories 54.

To facilitate determining the manipulated variable setpoint trajectories54, the control optimization circuitry 32 may utilize the process model28, constraints 56 on the process 12, and a control objective function58. As discussed above, the manipulated variable setpoint trajectories54 may include a manipulated variable for each time step over a controlhorizon. Accordingly, the control optimization circuitry 32 may use theprocess model 28 to determine a resulting controlled variable for eachtime each over the control horizon. Additionally, the controloptimization circuitry 32 may determine the manipulated variablessubject to constraints 56 on the manipulated variables and/or thecontrolled variables and based at least in part on the control objectivefunction 58, which describes weighting of various aspects of the process12 affected by the manipulated variables and/or controlled variables.

To help illustrate, one example of a mathematical representationoperations performed by the control optimization circuitry 32 isdescribed below. In some embodiments, the control objective function 58may be as follows:

$\begin{matrix}{J_{C} = {{\sum\limits_{j = 1}^{n_{y}}\;{\sum\limits_{i = 1}^{k}\;{W_{y,j}( \frac{{y_{j}( {t + i} )} - y_{j}^{T}}{R_{yj}} )}^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;{\sum\limits_{i = 1}^{k}\;{W_{{du},n}( \frac{\Delta\;{u_{n}( {t + i} )}}{R_{dun}} )}^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;{\sum\limits_{i = 1}^{k}\;{W_{u,n}( \frac{{u_{n}( {t + i} )} - u_{n}^{T}}{R_{un}} )}^{2}}}}} & (5)\end{matrix}$where J_(C) is the control objective function 58; n_(y) is the number ofcontrolled variables; n_(u) is the number of manipulated variables; k isthe number of time steps during the control horizon; y_(j)(t+i) a vectorof the controlled variables at time step t+i; y^(T) _(j) is a vector ofthe desired controlled variable trajectories 48; Δu_(n)(t+i) is a vectorof the change of the manipulated variables from time step t+i−1 to timestep t+i; u_(n)(t+i) is a vector of the manipulated variables at timestep t+i; u^(T) _(n) is a vector of the desired manipulated variabletrajectories 50; R_(yj), R_(dun), and R_(un) are scaling factors; andW_(y,j), W_(du,n), and W_(u,n) are tuning parameters 60.

More specifically, in some embodiments, the scaling factors R_(yj),R_(dun), and R_(un) may be determined based at least in part on range ofthe manipulated variable and/or range of the controlled variable.Additionally, as discussed above, the desired controlled variabletrajectory 48 and the desired manipulated variable trajectory 50 may bedetermined, for example, by the state transition optimization circuitry30 and/or be a user input. Furthermore, the value of the controlledvariable at time t+i may be determined using the process model 28. Forexample, when the process model 28 is a parametric hybrid model 28A, thecontrolled variable at time t+i may be determined as follows:y _(j)(t+i)=G(u _(n)(t+i),x _(t+i),θ_(t+1))  (6)where y_(j)(t+i) the vector of the controlled variables at time stept+i; u_(n)(t+i) is the vector of the manipulated variables at time stept+i; x_(t+i) is a vector of the state variables at time step t+i andθ_(t+i) is a vector of fundamental model parameters at time step t+i. Inother words, the process model 28 may facilitate determining theresulting controlled variable at a time step in the control horizonbased at least in part on the expected operating state of the process 12at that time step.

Additionally, as discussed above, the process 12 may have constraints 56on the manipulated variables and/or the controlled variables. In someembodiments, the constraints 56 may be as follows:Δu _(−,n) ≤Δu _(n)(t+i)≤Δu _(+,n)  (7)u _(min,n) ≤u _(n)(t+i)≤u _(max,n)  (8)where Δu_(−,n) is a vector of minimum rate of changes for themanipulated variables; Δu_(+,n) is a vector of maximum rate of changesfor the manipulated variables; u_(min,n) is a vector of minimum valuesfor the manipulated variables; u_(max,n) is a vector of maximum valuesfor the manipulated variables; Δu_(n)(t+i) is the vector of the changeof the manipulated variables from time step t+i−1 to time step t+i; andu_(n)(t+i) is the vector of the manipulated variables at time step t+i.In other words, equation (7) describes a constraint 56 on rate of changeof the manipulated variable and equation (8) describes a constraint 56on value of the manipulated variable. Additionally or alternatively, theprocess 12 may include constraints 56 on a rate of change of thecontrolled variables and/or constraints 56 on value of the controlledvariables.

Thus, as described above, the model predictive control system 14A mayuse the control optimization circuitry 32 to determine manipulatedvariable setpoints 54, which when implemented control operation of theprocess 12. One embodiment of a control process 62 for operating thecontrol optimization circuitry 32 is described in FIG. 6. Generally, thecontrol process 62 includes determining a current operating state(process block 64), determining a desired operating state (process block66), determining a desired state transition trajectory (process block68), determining manipulated variable setpoint trajectories (processblock 70), and implementing manipulated variable setpoints (processblock 72). In some embodiments, the control process 62 may beimplemented by instructions stored in a tangible, non-transitory,computer-readable medium, such as memory 26, executed by processingcircuitry, such as processor 24.

Accordingly, the control system 14 may determine a current operatingstate of the process 12 (process block 64). As described above, thecurrent operating state of the process 12 may include manipulatedvariables 44 and controlled variables 46 of the process determinedduring a present time step. Thus, in some embodiments, the controlsystem 14 may determine the current operating state of the process 12 bypolling one or more of the sensors 22 that measure operationalparameters of the process 12.

Based at least in part on the measured operational parameters thecontrol system 14 may determine the current operating state of theprocess 12. For example, in some embodiments, the manipulated variables44 and/or the controlled variables 46 may be the operational parametersmeasured by the sensors 22. However, in other embodiments, one or moreof the manipulated variables 44 and/or the controlled variables 46 maynot be directly measured. Accordingly, in such embodiments, the controlsystem 14 may determine the current operating state using a processmodel 28 that describes the relationship between the measuredoperational parameters and the one or more of the manipulated variables44 and/or the controlled variables 46

Additionally, the control system 14 may determine a desired operatingstate 42 of the process 12 (process block 66). Generally, the desiredoperating state 42 may be determined by user inputs and/or based onother operations of the control system 14. Depending on theimplementation, the desired operating state 42 may be defined withvarying specificity. For example, in some embodiments, the desiredoperating state 42 may describe the operating state the process 12 isdesired to be at after a control horizon (e.g., one or more future timesteps) as well as the desired state transition trajectory to transitionthe process 12 from the current operating state to the desired operatingstate 42. Thus, in such embodiments, the desired state transitiontrajectory is generally based on skill/knowledge of the user (e.g.,control engineer).

To reduce the reliance of user skill/knowledge, in some embodiments, thedesired operating state 42 may merely describe the operating state theprocess 12 is desired to be at after the control horizon. Thus, in suchembodiments, the control system 14 may systematically determine thedesired state transition trajectory based on the desired operating state42 (process block 68). As described above, the control system 14 maydetermine the desired state transition trajectory (e.g., desiredcontrolled variable trajectories 48 and desired manipulated variabletrajectories 50) using the state transition optimization circuitry 30.

More specifically, the control system 14 may utilize the process model28 to determine various sets of manipulated variable and controlledvariable trajectories that are expected to transition the process 14from the current operating state to the desired operating state 42. Thecontrol system 14 may then identify a set of the manipulated variableand controlled variable trajectories as the desired control variabletrajectories 48 and the desired manipulated variable trajectories 50based at least in part on the transition objective function 52. Asdescribed above, the transition objective function 52 may describe costsassociated with the value of the manipulated variables and controlledvariables at each time step over the control horizon and costsassociated with changes to the manipulated variables and controlledvariables between each time step, or any combination thereof. As such,the control system 14 may determine the desired state transitiontrajectory by selecting the desired control variable trajectories 48 andthe desired manipulated variable trajectories 50 with a lowestassociated cost.

The control system 14 may then determine the manipulated variablesetpoint trajectories 54 using the control optimization circuitry 32(process block 70). As described above, the control system 14 maydetermine the manipulated variable setpoint trajectories 54 based atleast in part on the desired operating trajectory (e.g., the desiredcontrol variable trajectories 48 and the desired manipulated variabletrajectories 50 over a control horizon) and the current operating stateof the process 12 (e.g., manipulated variables 44 and controlledvariables 46 determined for the present time step).

More specifically, the control system 14 may determine variousmanipulated variable trajectories that satisfy any constraints 56 on themanipulated variables. Additionally, using the process model 28, thecontrol system 14 may determine resulting controlled variabletrajectories for each of the various manipulated variable trajectories.Furthermore, the control system 14 may identify the resulting controlledvariable trajectories that satisfy any constraints 56 on the controlledvariables. In this manner, the control system 14 may determine sets offeasible controlled variable trajectories and manipulated variabletrajectories that are expected to satisfy constraints 56 on the process12.

From the sets of feasible controlled variable trajectories andmanipulated variable trajectories, the control system 14 may identifythe manipulated variable setpoint trajectories 54 based at least in parton the control objective function 58. As described above, the controlobjective function 58 may contain weight factors of the process 12associated with the manipulated variables and/or controlled variables.For example, the control objective function 58 may define a costassociated with deviation of the controlled variables from the desiredcontrolled variable trajectory 48 over the control horizon, a costassociated with adjustments to the manipulated variables over thecontrol horizon, and a cost associated with deviation of the manipulatedvariables from the desired manipulated variable trajectories 50 over thecontrol horizon.

Accordingly, in some embodiments, the control system 14 may determinethe manipulated variable setpoint trajectories 54 by identifying the setof feasible trajectories with the least associated cost. The controlsystem 14 may then instruct automation components 20 to implementmanipulated variable setpoints from the manipulated variable setpointtrajectories 54 (process block 72). More specifically, the controlsystem 14 may instruct the automation components 20 to implement a setof manipulated variable setpoints from the manipulated variablesetpoints trajectories 54 corresponding with the time step whenimplemented.

As discussed above, the control objective function 58 may include tuningparameters 60 to facilitate determining the manipulated variablesetpoints trajectories 54. More specifically, the tuning parameters 60may provide a weighting between the various aspects of the process 12.For example, tuning parameters 60 may provide a weighting between thecost associated with deviation of the controlled variables from thedesired controlled variable trajectory 48 over the control horizon, thecost associated with adjustments to the manipulated variables over thecontrol horizon, and the cost associated with deviation of themanipulated variables from the desired manipulated variable trajectories50 over the control horizon.

Generally, the tuning parameters 60 may depend on properties of theprocess 12, the process model 28 used to describe the process 12, anddesired behavior of the process 12, for example, as defined by a user.As such, the tuning parameters 60 may be tuned after deployment of thecontrol system 14 to control the process 12 and/or periodicallythereafter, for example, when the desired behavior of the process 12 isadjusted. As described above, it may be possible to determine the tuningparameters 60 heuristically by iteratively running the process 12through a training process. However, such heuristic techniques aregenerally reliant on user (e.g., control engineer) skill knowledge ofthe process 12, the process model 28, and the control system 14 andlimited to offline applications.

To reduce reliance of user knowledge and/or enable online application, asystematic approach to determining the tuning parameters 60 may beemployed. One such approach utilizes the tuning optimization circuitry34 to determine the tuning parameters 60. In fact, utilizing the tuningoptimization circuitry 34 may enable determining tuning parameters 60for various processes 12, various process models 28, various controlsystems 14, or any combination thereof. In particular, the tuningoptimization circuitry 34 may be beneficial when periodically adjustingthe tuning parameters 60 after deployment, for example, by obviatingpausing operation to run a training sequence.

To help illustrate, one embodiment of the tuning optimization circuitry34 is described in FIG. 7. As depicted, the tuning optimizationcircuitry 34 receives the desired operating trajectory, which includesthe desired control variable trajectories 48 and the desired manipulatedvariable trajectories 50 over a control horizon, and the currentoperating state, which includes the manipulated variables 44 andcontrolled variables 46 determined for a current time step.Additionally, the tuning optimization circuitry 34 outputs the tuningparameters 60.

To facilitate determining the tuning parameters 60, the tuningoptimization circuitry 34 may utilize the process model 28, an augmentedobjective function 74 with soft constraints 76, a closed form solution78, and a tuning objective function 80. More specifically, the augmentedobjective function 74 may include the control objective function 58 andsoft constraints 76, which are determined based at least in part on theconstraints 56. Additionally, the closed form solution 78 may be asolution to the augmented objective function 74 as a function of thetuning parameters 60. Furthermore, the tuning objective function 80 maybe determined based at least in part on the control objective function58 and the closed form solution 78 such that the tuning objectivefunction 80 is a function of the tuning parameters 60. Thus, by solvingthe tuning objective function 80, the tuning optimization circuitry 34may determine the tuning parameters 60.

To help illustrate, one embodiment of a tuning process 82 using thetuning optimization circuitry 34 is described in FIG. 8. Generally, thetuning process 82 includes determining soft constraints (process block84), determining an augmented objective function (process block 86),determining a closed form solution (process block 88), determining atuning objective function (process block 90), determining tuningparameters (process block 92), and optionally determining robustness ofthe tuning parameters (process block 93). In some embodiments, thetuning process 82 may be implemented by instructions stored in atangible, non-transitory, computer-readable medium, such as memory 26,executed by processing circuitry, such as processor 24.

Accordingly, the control system 14 may determine the soft constraints 76(process block 84). More specifically, in some embodiments, the controlsystem 14 may determine the soft constraints 76 by transforminginequalities in the constraints 56 into equality constraints byintroducing additional variables. Various techniques may be utilized totransform the constraints 56 into soft constraints 76, such as penaltytechniques, augmented Lagrangian techniques, slack variable techniques,barrier function techniques, and the like.

For example, in some embodiments, the control system 14 may determinethe soft constraints 76 by introducing slack variables into theconstraints 56 (process block 94). More specifically, the slackvariables may facilitate transforming inequalities in the constraints 56into equality constraints in the soft constraints 76. For example, theconstraints 56 on rate of change of the manipulated variables describedin equation (7) may be transformed into soft constraints 76 as follows:Δu _(n)(t+i)−Δu _(−,n)−δ_(Δun,1)=0; and  (9)Δu _(+,n) −Δu _(n)(t+i)−δΔ_(un,2)=0  (10)subject to:δ_(Δun,1)≥0; and  (11)δ_(Δun,2)≥0  (12)where δ_(Δun,1) is a first rate of change slack variable; δ_(Δun,2) is asecond rate of change slack variable; Δu_(n)(t+i) is the a vector of thechange of the manipulated variables from time step t+i−1 to time stept+i; Δu_(−,n) is the vector of minimum rate of changes for themanipulated variables; and Δu_(+,n) is the vector of maximum rate ofchanges for the manipulated variables. Additionally, the constraints 56on value of the manipulated variables described in equation (8) may betransformed into soft constraints 76 as follows:u _(n)(t+i)−u _(min,n)−δ_(un,1)=0; and  (13)u _(max,n) −u _(n)(t+i)−δ_(un,2)=0  (14)subject to:δ_(un,1)≥0; and  (15)δ_(un,2)≥0  (16)where δ_(un,1) is a first value slack variable; δ_(un,2) is a secondvalue slack variable; u_(n)(t+i) is the vector of the manipulatedvariables at time step t+i; u_(min,n) is the vector of minimum valuesfor the manipulated variables; and u_(max,n) is the vector of maximumvalues for the manipulated variables. Similarly, other constraints 56 onthe manipulated variables and/or controlled variables may be transformedinto soft constraints 76 by introducing slack variables.

In other embodiments, the soft constraints 76 may be determined bytransforming the constraints 56 into barrier functions (process block96). More specifically, the barrier functions may facilitatetransforming inequalities in the constraints 56 into equalityconstraints in the soft constraints 76, for example, by penalizingexceeding the constraints 56. For example, the constraints 56 on rate ofchange of the manipulated variables described in equation (7) may betransformed into soft constraints 76 as follows:

$\begin{matrix}{{g_{1}( {{\Delta\;{u_{n}( {t + i} )}},{\Delta\; u_{- {,n}}}} )} = \{ \begin{matrix}{\sum\limits_{i = 1}^{n_{u}}\;{- {\ln( {{\Delta\;{u_{n}( {t + i} )}} - {\Delta\; u_{- {,n}}}} )}}} & {{{{for}\mspace{14mu}\Delta\;{u_{n}( {t + i} )}} > {\Delta\; u_{- {,n}}}};{and}} \\\infty & {otherwise}\end{matrix} } & (17) \\{{g_{2}( {{\Delta\;{u_{n}( {t + i} )}},{\Delta\; u_{+ {,n}}}} )} = \{ \begin{matrix}{\sum\limits_{i = 1}^{n_{u}}\;{- {\ln( {{\Delta\; u_{+ {,n}}} - {\Delta\;{u_{n}( {t + i} )}}} )}}} & {{{for}\mspace{14mu}\Delta\;{u_{n}( {t + i} )}} < {\Delta\; u_{+ {,n}}}} \\\infty & {otherwise}\end{matrix} } & (18)\end{matrix}$where g₁(Δu_(n)(t+i), Δu_(−,n)) is a first rate of change barrierfunction; g₂(Δu_(n)(t+i), Δu_(+,n)) is a second rate of change barrierfunction; Δu_(n)(t+i) is the vector of the change of the manipulatedvariables from time step t+i−1 to time step t+i; Δu_(−,n) is the vectorof minimum rate of changes for the manipulated variables; and Δu_(+,n)is the vector of maximum rate of changes for the manipulated variables.Additionally, the constraints 56 on value of the manipulated variablesdescribed in equation (8) may be transformed into soft constraints 76 asfollows:

$\begin{matrix}{{g_{3}( {{u_{n}( {t + i} )},u_{\min,n}} )} = \{ \begin{matrix}{\sum\limits_{i = 1}^{n_{u}}\;{- {\ln( {{u_{n}( {t + i} )} - u_{\min,n}} )}}} & {{{{for}\mspace{14mu}{u_{n}( {t + i} )}} > {\Delta\; u_{\min,n}}};{and}} \\\infty & {otherwise}\end{matrix} } & (19) \\{{g_{4}( {{u_{n}( {t + i} )},u_{\max,n}} )} = \{ \begin{matrix}{\sum\limits_{i = 1}^{n_{u}}\;{- {\ln( {u_{\max,n} - {\Delta\;{u_{n}( {t + i} )}}} )}}} & {{{for}\mspace{14mu}{u_{n}( {t + i} )}} < {\Delta\; u_{\max,n}}} \\\infty & {otherwise}\end{matrix} } & (20)\end{matrix}$where g₃(u_(n)(t+i), u_(min,n)) is a first value barrier function:g₄(u_(n)(t+i), u_(max,n)) is a second value barrier function; u_(n)(t+i)is the vector of the manipulated variables at time step t+i; u_(min,n)is the vector of minimum values for the manipulated variables; andu_(max,n) is the a vector of maximum values for the manipulatedvariables. Similarly, other constraints 56 on the manipulated variablesand/or controlled variables may be transformed into soft constraints 76as barrier functions.

Based on the soft constraints 76, the control system 14 may determinethe augmented objective function 74 (process block 86). Morespecifically, the control system 14 may determine the augmentedobjective function 74 by augmenting the control objective function 58 toinclude the soft constraints 76. For example, the augmented objectivefunction 74 may include the terms of the control objective function 58plus each of the soft constraints 76.

For example, in embodiments where the soft constraints 76 are generatedusing slack variables, the augmented objective function 74 may be asfollows:J _(A) =J _(C) +[Δu _(n)(t+i)−Δu _(−,n)−δ_(Δun,1) ]+[Δu _(+,n) −Δu_(n)(t+i)−δ_(Δun,2) ]+[u _(n)(t+i)−u _(min,n)−δ_(un,1) ]+[u _(max,n) −u_(n)(t+i)−δ_(un,1)]  (21)subject to:δΔ_(un,1)≥0;  (22)δΔ_(un,2)≥0;  (23)δ_(un,1)≥0; and  (24)δ_(un,2)≥0  (25)where J_(A) is the augmented objective function 74; J_(C) is the controlobjective function 58; δ_(Δun,1) is the first rate of change slackvariable; δ_(Δun,2) is the second rate of change slack variable;Δu_(n)(t+i) is the vector of the change of the manipulated variablesfrom time step t+i−1 to time step t+i; Δu_(−,n) is the vector of minimumrate of changes for the manipulated variables; Δu_(+,n) is the vector ofmaximum rate of changes for the manipulated variables; δ_(un,1) is thefirst value slack variable; δ_(un,2) is the second value slack variable;u_(n)(t+i) is the vector of the manipulated variables at time step t+i;u_(min,n) is the vector of minimum values for the manipulated variables;and u_(max,n) is the vector of maximum values for the manipulatedvariables.

Additionally, in embodiments where soft constraints 76 are barrierfunctions, the augmented objective function 74 may be as follows:J _(A) =J _(C) +r ₁ *g ₁(Δu _(n)(t+i),Δu _(−,n))+r ₂ *g ₂(Δu_(n)(t+i),Δu _(+,n))+r ₃ *g ₃(u _(n)(t+i),u _(min,n))+r ₄ *g ₄(u_(n)(t+i),u _(max,n))  (26)subject to:r ₁≥0;  (27)r ₂≥0;  (28)r ₃≥0; and  (29)r ₄≥0  (30)where J_(A) is the augmented objective function 74; J_(C) is the controlobjective function 58; g₁(Δu_(n)(t+i), Δu_(−,n)) is the first rate ofchange barrier function; g₂(Δu_(n)(t+i), Δu_(+,n)) is the second rate ofchange barrier function; g₃(u_(n)(t+i), u_(min,n)) is the first valuebarrier function; g₄(u_(n)(t+i), u_(max,n)) is the second value barrierfunction; and r₁, r₂, r₃, r₄ are barrier function variables.

Once determined, the control system 14 may determine a closed formsolution 78 to the augmented objective function 74 (process block 88).More specifically, the closed form solution 78 may be determined over aset control horizon based on the current operating state and the desiredoperating state 42. As such, most of the variables in the augmentedobjective function 74 may be known, thereby enabling the augmentedobjective function 74 to be solved in a deterministic manner.

For example, in the above described augmented objective function 74, thenumber of controlled variables, n_(y), the number of time steps duringthe control horizon, k, the scaling factors, R_(yj), R_(dun), andR_(un), the of minimum rate of changes for the manipulated variables,Δu_(−,n), the of maximum rate of changes for the manipulated variables,Δu_(+,n), the maximum values for the manipulated variables, u_(max,n),and the minimum values for the manipulated variables, u_(min,n), may beknown. Additionally, the desired controlled variable trajectories 48,y^(T) _(j), and the desired manipulated variable trajectories 50, u^(T)_(n), may be determined by the state transition optimization circuitry30 based at least in part on the current operating state (e.g.,manipulated variables 44 and controlled variables 46) and the desiredoperating state 42. Furthermore, the controlled variables at each timestep, y_(j)(t+i), manipulated variables at each time step, u_(n)(t+i),and the rate of change of the manipulated variables at each time step,Δu_(n)(t+i), may be determined using the process model 28 based at leastin part on the current operating state and the desired operating state42.

In this manner, the control system 14 may determine the closed formsolution 78 (e.g., manipulated variable trajectories over the controlhorizon), for example, by minimizing the augmented objective function74. In embodiments where the soft constraints 76 are generated usingslack variables, the closed form solution 78 may be as follows:u _(cLoop) =A(W _(y,j) ,W _(du,n) ,W_(u,n),δ_(Δun,1),δ_(Δun,2),δ_(un,1),δ_(un,1))  (31)where u_(cLoop) is a vector the closed loop solution 78; W_(y,j) is afirst tuning parameter; W_(du,n) is a second tuning parameter, W_(u,n)is a third tuning parameter; δ_(Δun,1) is the first rate of change slackvariable; δ_(Δun,2) is the second rate of change slack variable;δ_(un,1) is the first value slack variable; and δ_(un,2) is the secondvalue slack variable.

Additionally, in embodiments where the soft constraints 76 are definedas barrier functions, the closed form solution 78 may be as follows:u _(cLoop) =B(W _(y,j) ,W _(du,n) ,W _(u,n) ,r ₁ ,r ₂ ,r ₃ ,r ₄)  (32)where u_(cLoop) is the vector the closed loop solution 78; W_(y,j) isthe first tuning parameter; W_(du,n) is the second tuning parameter; andW_(u,n) is the third tuning parameter; r₁ is a first barrier functionvariable; r₂ is a second barrier function variable; r₃ is a thirdbarrier function variable; and r₄ is a fourth barrier function variable.

The control system 14 may determine the tuning objective function 80(process block 90). In some embodiments, the tuning objective function80 may be determined by augmenting one or more of the terms in thecontrol objective function 58. For example, when in embodiments wherethe soft constraints 76 are generated using slack variables, the tuningobjective function 80 may be as follows:

$\begin{matrix}{{{J_{T} = {{\sum\limits_{j = 1}^{n_{y}}\;{\sum\limits_{i = 1}^{k}\;{W_{y,j}( \frac{{y_{j}( {t + i} )} - y_{j}^{T}}{R_{yj}} )}^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;( \delta_{{\Delta\;{un}},1} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( \delta_{{un},2} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( \delta_{{un},1} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( \delta_{{un},2} )^{2}}}}\mspace{20mu}{{subject}\mspace{14mu}{to}\text{:}}}\mspace{11mu}} & (33) \\{\mspace{79mu}{{\delta_{{\Delta\;{un}},1} \geq 0};}} & (34) \\{\mspace{79mu}{{\delta_{{\Delta\;{un}},2} \geq 0};}} & (35) \\{\mspace{79mu}{{\delta_{{un},1} \geq 0};{and}}} & (36) \\{\mspace{79mu}{\delta_{{un},2} \geq 0}} & (37)\end{matrix}$where J_(T) is the tuning objective function 80; n_(y) is the number ofcontrolled variables; n_(u) is the number of manipulated variables; k isthe number of time steps during the control horizon; y_(j)(t+i) thevector of the controlled variables at time step t+i; y^(T) _(j) is avector of the desired controlled variable trajectories 48; R_(yj) is ascaling factor; δ_(Δun,1) is the first rate of change slack variable;δ_(Δun,2) is the second rate of change slack variable; δ_(un,1) is thefirst value slack variable; and δ_(un,2) is the second value slackvariable.

Additionally, in embodiments where the soft constraints 76 are generatedas barrier functions, the tuning objective function 80 may be asfollows:

$\begin{matrix}{{{J_{T} = {{\sum\limits_{j = 1}^{n_{y}}\;{\sum\limits_{i = 1}^{k}\;{W_{y,j}( \frac{{y_{j}( {t + i} )} - y_{j}^{T}}{R_{yj}} )}^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;( r_{1} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( r_{2} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( r_{3} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( r_{4} )^{2}}}}\mspace{20mu}{{subject}\mspace{14mu}{to}\text{:}}}\mspace{11mu}} & (38) \\{\mspace{79mu}{{r_{1} \geq 0};}} & (39) \\{\mspace{79mu}{{r_{2} \geq 0};}} & (40) \\{\mspace{79mu}{{r_{3} \geq 0};{and}}} & (41) \\{\mspace{79mu}{r_{4} \geq 0}} & (42)\end{matrix}$where J_(T) is the tuning objective function 80; n_(y) is the number ofcontrolled variables; n_(u) is the number of manipulated variables; k isthe number of time steps during the control horizon; y_(j)(t+i) thevector of the controlled variables at time step t+i; y^(T) _(j) is thevector of the desired controlled variable trajectories 48; R_(yj) is ascaling factor; r₁ is the first barrier function variable; r₂ is thesecond barrier function variable; r₃ is the third barrier functionvariable; and r₄ is the fourth barrier function variable.

Additionally, the control system 14 may define the controlled variablesat each time step based at least in part on the closed form solution 78.For example, in some embodiments, the controlled variables at each timestep may be defined as follows:Y _(j)(t+i)=∧_(j)(u _(cLoop))  (43)where y_(j)(t+i) the vector of the controlled variables at time step t+iand u_(cLoop) is a vector of the closed loop solution 78. In thismanner, the tuning objective function 80 may be defined independent fromthe manipulated variables and based at least in part on the tuningparameters, which parameterize the closed form solution 78. Thisparametrization may facilitate systematically solve the tuning objectivefunction to tune the control system 14. In particular, sensitivityanalysis may be applied to the closed form solution to determineparameter ranges, in which manipulated variables of the control system14 are less sensitive (e.g., fragile) to variation in tuning parameters.To facilitate, the optimization problem for tuning may be constrainedaway from the regions of high sensitivity. In some embodiments, thesensitivity may be defined as the ratio of a change in a manipulatedvariable over a change in a tuning parameter. In such embodiments, themanipulated variable may be overly sensitive when sensitivity is greaterthan a sensitivity threshold and sufficiently insensitive when notgreater than the sensitivity threshold.

Thus, using the tuning objective function 80, the control system 14 maydetermine the tuning parameters 60 via explicit optimization (processblock 92). For example, in some embodiments, the control system 14 maydetermine a set of tuning parameters 60 that minimizes the tuningobjective function 80. As described above, the determined tuningparameters 60 may then be utilized in the control objective function 58,thereby enabling the control system 14 to execute control optimizationcircuitry 32 and control operation of the process 12. Thus, the tuningparameters 60 may substantially affect operation of the process 12.

As such, in some embodiments, the control system 14 may determinerobustness of the tuning parameters 60 before implementation (processblock 93). More specifically, since the tuning parameters 60 may bedetermined online (e.g., while control system 14 is controllingoperation of the process 12), it may be undesirable for changes intuning parameters to cause drastic changes in operating condition of theprocess 12.

Accordingly, in some embodiments, the control system 14 may determinethe robustness based at least in part on the effects implementing thetuning parameters 60 is expected to have on the operation of the process12. For example, the control system 14 may substitute the determinedtuning parameters 60 back into the augmented objective function 80. Bysolving the augmented objective function 80, the control system 14 maydetermine amount of disturbance implementing the tuning parameters 60 isexpected to cause in the process 12.

When the amount of disturbance is undesirable, the control system 14 mayre-determine the tuning parameters 60 with reduced sensitivity to theclosed form solution 78. For example, the control system 14 mayre-determine tuning parameters 60 such that the tuning parameter 60 forcost associated with deviation of the controlled variable from thedesired controlled variable trajectory 48 is reduced. In this manner,optimal tuning may be traded off against amount of disturbance to theprocess 12, which is particularly useful when implemented online.

To further facilitate determining the tuning parameters 60 online,computational efficiency may be improved. It is appreciated thatcomputational efficiency may be dependent on complexity of the process12 and/or the process model 28. In fact, in some instances, matricesused in the state transition optimization circuitry 30, the controloptimization circuitry 32, and/or the tuning optimization circuitry 34may be large (e.g., a 100×100 matrix). As such, computations with suchlarge matrices may be complex, for example, to determine the inverse ofthe matrices. In some embodiments, the computational complexity may bereduced by using rank constraints, which enables reducing the size ofthe matrices. For example, the matrices may be mapped to a differentvector space based on importance with the first column being the mostimportant, the second column being the second most important, and so on.In fact, this may enable selecting the portion (e.g., 10×10) of thelarge matrices determined as most important as an approximation of thelarge matrices. In this manner computational complexity may be reducedand computational efficiency improved.

Additional Details

The present disclosure provides optimization-based tuning (OBT)processes for control systems where at least one of a process model,constraints, and an objective function is a parametric hybrid model. Insome embodiments, the parametric hybrid model may be linear ornonlinear. Additionally, the control system may determine at least onetuning parameter for the control system based on solving an explicitoptimization problem. In fact, optimization-based tuning (OBT) processesmay be equally applicable to offline and online deployment.

More specifically, the optimization-based tuning (OBT) processes enablea completely systematic methodology for tuning of multi-inputmulti-output (MIMO) linear and nonlinear optimization-based control(OBC) systems. Systematic tuning of the OBC systems has proven achallenge especially for MIMO processes with significant non-diagonaldynamics. Most of existing tuning strategies are based on heuristics andtherefore rely heavily on the expertise of the control engineer. Assuch, heuristic tuning has generally been limited to exclusively offlineuse. Comparatively, the optimization-based tuning (OBT) processes enableonline deployment with no additional input from the user to determinethe tuning parameters.

Additionally the tuning parameters may be robustly determined by takinginto account uncertainty in a process model, which may be due tounmeasured disturbances and/or parametric uncertainty in process model.Thus, the tuning parameters may be determined along with an uncertaintyenvelope.

Furthermore, the optimization problem for determining the tuningparameters of the control system are based on the result of a secondoptimization problem, such as a steady-state optimization problem givencontrolled variable targets. In fact, the optimization-based tuning(OBT) processes accommodate user-defined controlled variable priorityrankings, where not all of the controlled variable targets can be metgiven available degrees of freedom. In this manner, robustness of thetuning parameters may be improved based on the user-defined priority.For example, a slack variable may be introduced so that a constrainedoptimization can achieve the tuning that will respect the user-definedranking.

One embodiment of an optimization-based tuning (OBT) process isdescribed as follows:

Utilization of Parametric Hybrid Models: PHM will be used as themodeling framework for the process. The parameterized nature of theprocess model enables automatic tuning for nonlinear processes.Furthermore, robust online deployment of the tuning algorithm may beenabled with parametric hybrid models in a manner that is easilyunderstood/manageable by the operators. A detailed description ofparametric hybrid models of interest is provided in U.S. Pat. No.8,019,701, entitled “TRAINING A MODEL OF A NON-LINEAR PROCESS,” which isincorporated herein by reference in its entirety.

Determination of Optimal Trajectory for State Transition: Anoptimization problem will be formulated whose solution is the optimaltrajectory for state transition from its current operating state to adesired operating state defined by the user or programmaticallygenerated to cover the feasible operation space for the process. Given acandidate path for state transition, the parametric hybrid modelgoverning process behavior along that transition path may be defined.

Formation of an Augmented Performance Objective Function: A typicalproblem formulation for an MPC problem is a constrained quadraticprogram such as:

$J = {{{\sum\limits_{j = 1}^{n_{y}}\;{\sum\limits_{i = 1}^{p}\;{W_{y,j}( \frac{{y_{j}( {t + i} )} - y_{j}^{t}}{R_{yj}} )}^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;{\sum\limits_{i = 1}^{p}\;{W_{{du},n}( \frac{\Delta\;{u_{n}( {t + i} )}}{R_{dun}} )}^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;{\sum\limits_{i = 1}^{p}\;{W_{u,n}( \frac{{u_{n}( {t + i} )} - u_{n}^{t}}{R_{un}} )}^{2}}}\mspace{20mu} - {\Delta\; u_{- {,n}}}} \leq {\Delta\;{u_{n}( {t + i} )}} \leq {\Delta\; u_{+ {,n}}}}$  u_(min , n) ≤ u_(n)(t + i) ≤ u_(max , n)  y_(j)(t + i) = Φ_(j)(u, y_(j), p, i)where W_(y,j) is a tuning parameter denoting the significance/cost ofdeviation of the j-th output, y_(j)(t+i), from the desired profile forthe j-th output, y^(t) _(j); W_(du,n) is a tuning parameter denoting thesignificance/cost of changes to the n-th output, Δu_(n)(t+i); W_(u,n) isa tuning parameter denoting the significance/cost of deviation of then-th input, u_(n)(t+i), from the desired profile for the n-th input,u_(n) ^(t); index “i” refers to the samples of the j-th output over theprediction horizon; R_(yj) is a scaling factor; R_(dun) is a scalingfactor; and R_(un) is a scaling factor. Additionally,−Δu_(−,n)≤Δu_(n)(t+i)≤Δu_(+,n) is the constraint set for the rate ofchange for the input variables, u_(min,n)≤u_(n)(t+i)≤u_(max,n) is theconstraint set for the input values, and y_(j)(t+i)=Φ_(j)(u, y_(t), p,i)is an update law for the j-th process output at time instance (t+i) as afunction of process input vector u, past output vector y_(t), and modelparameter vector p. The output update function here is just one exampleof numerous ways the evolution of a dynamic system.

It may be possible to construct a closed form solution to theunconstrained optimization problem where a feedback law is created as afunction of tuning parameters. The closed-form solution for theunconstrained optimization problem however often violates theoperational constraints resulting in suboptimal tuning of the MPCcontroller. Furthermore, the unconstrained closed-form solution isfrequently ill-conditioned (especially as the number of inputs/outputsincreases).

The present disclosure proposes the formation of an augmented objectivefunction where explicit constraints on input variables are transformedinto soft constraints in the form of a penalty term in the augmentedobjective function.

The augmented objective function can be formed in a variety of ways. Onecommonly used approaches includes introducing a slack variable totransform the inequality constraint into an equality constraint and thenaugment the objective function with the 2-norm of the equalityconstraint treating the slack variable as an additional decisionvariable. More specifically, the inequality constraintu_(min,n)≤u_(n)(t+i)≤u_(max,n) can be rewritten as two equalityconstraints with the help of two slack variables as follows:u _(n)(t+i)−u _(min,n)−δ_(un,1)=0u _(max,n) −u _(n)(t+i)−δ_(un,2)=0Another commonly used approach includes using barrier function (mostcommonly used barrier function is the natural logarithm) to augment theobjective function with the constraints.

Derivation of a Closed-Form Solution to the Augmented ObjectiveFunction: The closed form solution is calculated over the anticipatedstate transition trajectory accounting for the variation in parametrichybrid models as a function of process operating condition. Theclosed-form unconstrained solution for the augmented cost function willbe a function of the tuning parameters as well as the parametersintroduced in the course of augmenting the original performanceobjective function J, for example δ_(un,1) and δ_(un,2). Moreover, usingparametric hybrid models may enable determination (e.g., calculation) ofthe closed form solution to the augmented optimization cost (e.g.,objective) function when the process is nonlinear and/or characteristicof the process (e.g., gain and/or time constants) vary over ananticipated control horizon.

In some embodiments, an optimization-based tuning system may determinetuning parameters of the control system online in real-time or nearreal-time and without the involvement of a human expert. In suchembodiments, number of constraints in the augmented objective functionmay be large, thereby reducing reliability of online deployment.Accordingly, to online deployment (e.g., in real-time or nearreal-time), constraints may be ranked when determining a closed formsolution to the augmented optimization objective function.

Definition of an Objective Function for the Optimal Tuning: Theobjective function for the optimal tuning may capture the desiredbehavior of the system under the control system (e.g. a model predictivecontrol system). Setting the control strategy to rank-constrainedclosed-form solution leaves the optimization problem for the optimaltuning a function of tuning coefficients and the slack/barrier parameterthat can be solved via explicit nonlinear optimization. An exampleobjective function will be as follows:

$J_{Tuning} = {{{\sum\limits_{j = 1}^{n_{y}}\;{\sum\limits_{i = 1}^{p}\;( \frac{{y_{j}( {t + i} )} - y_{j}^{d}}{R_{yj}} )^{2}}} + {\sum\limits_{n = 1}^{n_{u}}\;( \delta_{{un},1} )^{2}} + {\sum\limits_{n = 1}^{n_{u}}\;( \delta_{{un},2} )^{2}} - {\Delta\; u_{- {,n}}}} \leq {\Delta\;{u_{n}( {t + i} )}} \leq {\Delta\; u_{+ {,n}}}}$u_(min , n) ≤ u_(n)(t + i) ≤ u_(max , n) y_(j)(t + i) = Λ_(j)(U_(cloop))where U_(cloop) is the result of derivation in step 4 and is a functionof the tuning parameters for MPC (e.g., W_(y,j), W_(du,n), and W_(u,n)the slack variables).

Determination of Tuning Parameters by Solving the Optimization Problemfor Optimal Tuning: The optimization problem for optimal tuning is ingeneral a constrained nonlinear programming for which a robust solutionis sought. To ensure the robustness of the tuning parameters, in someembodiments, minimization of an objective function used for tuning, forexample J_(Tuning), may be traded off against sensitivity of therank-constrained closed-form solution, for example U_(cloop), to tuningparameters. This approach is of particular importance if/when theautomatic tuning algorithm is implemented online. Solving theoptimization problem for tuning with rank constraints while improves thenumerical properties of the search algorithm does not directly addressthe sensitivity of the MPC response to changes in tuning parametervalues.

Accordingly, the present disclosure provides technical effects thatinclude providing a systematic approach for tuning a control system thatcontrols a process. More specifically, tuning parameters used todetermine manipulated variable setpoints implemented in the process maybe determined by an explicit tuning optimization. In some embodiments,the tuning optimization may be based on a control objective function,current operating state of the process, a desired operating state of theprocess, a process model, and any constraints on the process. As such,the tuning parameters may be determined systematically for variousprocesses, control systems, and process models with reduced reliance onuser (e.g., control engineer) knowledge.

In some embodiments, the automatic tuning module 34 may be deployed onan industrially hardened central processing unit (CPU) that plugs intoan industrial controller enclosure. In such embodiments, process datamay be communicated to the automatic tuning module 34 through a fastdata communication line (e.g. the backplane) and the tuning parametersmay be communicated back to a controller that controls operation of aprocess via the fast data communication line. In this manner,determination of tuning parameters may be decoupled from control ofprocess (e.g., by the controller), thereby enabling execution atdifferent frequencies.

While only certain features of the disclosure have been illustrated anddescribed herein, many modifications and changes will occur to thoseskilled in the art. It is, therefore, to be understood that the appendedclaims are intended to cover all such modifications and changes as fallwithin the true spirit of the disclosure.

The invention claimed is:
 1. An industrial automation system comprising:a motor configured to actuate a conveyer belt based at least in part amanipulated variable setpoint to facilitate performing an industrialautomation process; a sensor configured to measure an operationalparameter of the motor; and a control system communicatively coupled tothe motor and the sensor, wherein the control system comprises: memoryconfigured to store a process model that models operation of theindustrial automation process by relating one or more manipulatedvariables of the industrial automation process and a controlled variableof the industrial automation process; control optimization circuitryprogrammed to determine the manipulated variable setpoint to beimplemented by the motor based at least in part on the process model,the operational parameter of the motor, and a control objective functionsubject to constraints on value, rate of change, or both of themanipulated variable setpoint, wherein the control objective functioncomprises a tuning parameter that describes weighting between aspects ofthe industrial automation process affected by the manipulated variablesetpoint; and tuning optimization circuitry programmed to: convert theconstraints on the manipulated variable setpoint into soft constraints;determine an augmented objective function subject to a constraint on anintermediate variable associated with the soft constraints, wherein theaugmented objective function comprises the control objective functionand the intermediate variable associated with the soft constraints;determine a closed-form manipulated variable setpoint trajectory as afunction of the tuning parameter in the control objective function basedat least in part on the process model, the operational parameter of themotor, and the augmented objective function; determine a tuningobjective function subject to the constraint on the intermediatevariable associated with the soft constraints, wherein the tuningobjective function comprises the tuning parameter in the controlobjective function, the intermediate variable associated with the softconstraints, and the controlled variable of the industrial automationprocess; and determine the tuning parameter included in the controlobjective function based at least in part on the tuning objectivefunction and the controlled variable of the industrial automationprocess defined as a function of the closed-form manipulated variablesetpoint trajectory.
 2. The industrial automation system of claim 1,comprising state optimization circuitry programmed to determine adesired controlled variable trajectory and a desired manipulatedvariable trajectory over a control horizon based at least in part on theprocess model, a current operating state of the industrial automationprocess, and a desired operating state of the industrial automationprocess after the control horizon; wherein the control optimizationcircuitry is programmed to determine a manipulated variable setpointtrajectory based at least in part on the current operating state of theindustrial automation process, the desired controlled variabletrajectory, and the desired manipulated variable trajectory, wherein themanipulated variable setpoint trajectory comprises the manipulatedvariable setpoint at each of a plurality of time step during the controlhorizon.
 3. The industrial automation system of claim 1, wherein theoperational parameter of the motor comprises temperature of the motor,speed of the motor, or both.
 4. The industrial automation system ofclaim 1, wherein the process model comprises a parametric hybrid modelcomprising: an empirical model configured to determine empirical modeloutputs as a function of manipulated variables of the industrialautomation process and an empirical model parameter; a parameter modelconfigured to determine fundamental model parameters as a function ofthe manipulated variables and the empirical model outputs; and aparametric first-principles model configured to: determine current statevariables of the industrial automation process as a function of themanipulated variables, the fundamental model parameters, and previousstate variables of the industrial automation process; and determinecontrolled variables of the industrial automation process as a functionof the current state variables, the manipulated variables, and thefundamental model parameters.
 5. The industrial automation system ofclaim 1, wherein the soft constraints comprise the constraints on themanipulated variable setpoint expressed with a slack variable or as abarrier function to change inequality constraints into equalityconstraints.
 6. The industrial automation system of claim 1, wherein thetuning parameter describes cost associated with: value of themanipulated variable setpoint at each of a plurality of time stepsduring a control horizon; change of the manipulated variable setpointbetween each of the plurality of time steps; or both.
 7. The industrialautomation system of claim 1, wherein the tuning optimization circuitryis programmed to: determine a plurality of possible tuning parameters;and select one of the plurality of possible tuning parameters thatminimizes the tuning objective function as the tuning parameter.
 8. Theindustrial automation system of claim 1, wherein the industrialautomation system comprises an automation plant, a factory, or anycombination thereof.
 9. The industrial automation system of claim 1,wherein the control system comprises one or more model predictivecontrol (MPC) controllers, one or more proportional-integral-derivative(PID) controllers, one or more neural network controllers, one or morefuzzy logic controllers, or any combination thereof.
 10. A method forcontrolling operation of an industrial automation process, comprising:receiving, using a control system, an operational parameter of a motorthat actuates a conveyer belt to facilitate performing the industrialautomation process from a sensor; converting, using the control system,constraints on value, rate of change, or both of a manipulated variablesetpoint to be implemented by the motor into soft constraints;determining, using the control system, an augmented objective functionsubject to a constraint on an intermediate variable associated with thesoft constraints, wherein the augmented object function comprises acontrol objective function and the intermediate variable associated withthe soft constraints; determining, using the control system, aclosed-form manipulated variable setpoint trajectory as a function oftuning parameters in the control objective function based at least inpart on the operational parameter of the motor, the augmented objectivefunction, and a process model that models operation of the industrialautomation process by relating manipulated variables of the industrialautomation process and controlled variables of the industrial automationprocess; determining, using the control system, a tuning objectivefunction subject to the constraint on the intermediate variableassociated with the soft constraints, wherein the tuning objectivefunction comprises the tuning parameters in the control objectivefunction, the intermediate variable associated with the softconstraints, and the controlled variable of the industrial automationprocess; determining, using the control system, a first set of thetuning parameters included in the control objective function based atleast in part on the tuning objective function and the controlledvariable of the industrial automation process defined as a function ofthe closed-form manipulated variable setpoint trajectory; determining,using the control system, the manipulated variable setpoint to beimplemented by the motor based at least in part on the process model,the operational parameter of the motor, and the control objectivefunction, wherein the control objective function comprises the tuningparameters as weighting on aspects of the industrial automation processaffected by the manipulated variables of the industrial automationprocess, the controlled variables of the process, or both; andcontrolling, using the control system, operation of the industrialautomation process by instructing the motor to actuate the conveyer beltto implement the manipulated variable setpoint.
 11. The method of claim10, wherein converting the constraint on the manipulated variablesetpoint into the soft constraints comprises transforming inequalitiesin the constraints to equalities by: introducing slack variables intothe constraints such that the closed-form manipulated variable setpointtrajectory is a function of at least the slack variables; or determiningbarrier functions comprising barrier variables based at least in part onthe constraints such that the closed-form manipulated variable setpointtrajectory is a function of at least the barrier variables.
 12. Themethod of claim 10, comprising determining, using the control system,robustness of the tuning parameters before using the tuning parametersin the control objective function, wherein determining robustnesscomprises: determining effect on the industrial automation processexpected to be caused by implementing the first set of tuning parametersby solving the augmented objective function using the first set oftuning parameters; implementing the first set of tuning parameters inthe control objective function when the effect is less than a threshold;and determining a second set of tuning parameters when the effect is notless than the threshold such that the second set of tuning parameters isless sensitive than the first set to the closed-form manipulatedvariable setpoint trajectory.
 13. The method of claim 10, comprisingimplementing rank constraints on matrices used to determine the tuningparameters or the manipulated variable setpoint.
 14. The method of claim10, comprising: determining, using the control system, a plurality ofpossible manipulated variable setpoint trajectories that facilitatetransitioning the industrial automation process from a current operatingstate to a desired operating state after a control horizon subject tothe constraints on the manipulated variable setpoint to be implementedby the motor, wherein each of the plurality of possible manipulatedvariable setpoint trajectories comprises the manipulated variablesetpoint at each of a plurality of time steps over the control horizon;determining, using the control system, a cost associated with each ofthe plurality of manipulated variable setpoint trajectories based atleast in part on the control objective function; and selecting, usingthe control system, one of the plurality of possible manipulatedvariable setpoints trajectories that minimizes the control objectivefunction.
 15. The method of claim 10, comprising instructing, using thecontrol system, the motor to actuate the conveyer belt to implement themanipulated variable setpoint to facilitate transitioning the industrialautomation process form a current operating state to a desired operatingstate after a control horizon.
 16. The method of claim 10, wherein theindustrial automation process comprises a manufacturing process, asteady state process, a batch process, a chemical process, a materialhandling process, an energy utilizing process, an energy productionprocess, or any combination thereof.
 17. A tangible, non-transitory,computer-readable medium that stores instructions executable by at leastone processor of a control system, wherein the instructions compriseinstructions to: instruct, using the at least one processor, a sensor todetermine an operational parameter of a motor that actuates a conveyerbelt to facilitate performing an industrial automation process; convert,using the at least one processor, constraints on value, rate of change,or both of a manipulated variable setpoint to be implemented by themotor into soft constraints; determine, using the at least oneprocessor, an augmented objective function subject to a constraint on anintermediate variable associated with the soft constraints, wherein theaugmented objective function comprises a control objective function andthe intermediate variable associated with the soft constraint;determine, using the at least one processor, a closed-form manipulatedvariable setpoint trajectory as a function of tuning parameters in thecontrol objective function based at least in part on the operationalparameter of the motor, the augmented objective function, and a processmodel that models operation of the industrial automation process byrelating manipulated variables of the industrial automation process andcontrolled variables of the industrial automation process; determine,using the at least one processor, a tuning objective function, whereinthe tuning objective function comprises the control objective function,the intermediate variable associated with soft constraints, and thecontrolled variable of the industrial automation process; tune, usingthe at least one processor, the tuning parameters included in thecontrol objective function based at least in part on the tuningobjective function and the controlled variable of the industrialautomation process defined as a function of the closed-form manipulatedvariable setpoint trajectory; determine, using the at least oneprocessor, the manipulated variable setpoint to be implemented by themotor based at least in part on the process model, the operationalparameter of the motor, and the control objective function, wherein thecontrol objective function comprises the tuning parameters as weightingon aspects of the industrial automation process affected by themanipulated variables of the industrial automation process, thecontrolled variables of the process, or both; and instruct, using the atleast one processor, the motor to actuate conveyer belt in accordancewith the manipulated variable setpoint.
 18. The computer-readable mediumof claim 17, comprising instructions to determine, using the at leastone processor, the soft constraints by introducing slack variables intothe constraints on the manipulated variable setpoint or transforming theconstraints on the manipulated variable setpoint into barrier functionscomprising barrier variables.
 19. The computer-readable medium of claim17, comprising instructions to: determine a desired operating trajectoryof the industrial automation process over a control horizon thattransitions the industrial automation process from a current operatingstate to a desired operating state after the control horizon, whereinthe desired operating trajectory comprises a desired controlled variabletrajectory and a desired manipulated variable trajectory, wherein thedesired controlled variable trajectory comprises a desired value of acontrolled variable of the industrial automation process at each of aplurality time steps and the manipulated variable trajectory comprises adesired value of manipulated variables of the industrial automationprocess at each of the plurality of time steps; and determine an actualoperating trajectory of the process to implement in the industrialautomation process that minimize the control objective function, whereinthe actual operating trajectory comprises a manipulated variablesetpoint trajectory, wherein the manipulated variable setpointtrajectory comprises value of the manipulated variable setpoint to beimplemented by the motor at each of the plurality of time steps.
 20. Thecomputer-readable medium of claim 19, wherein the tuning parameterincluded in the control objective function are configured to: weight acost associated with deviation of the manipulated variable setpoint andthe desired value of the manipulated variable at each of the pluralityof time steps; and the second tuning parameter is configured to describea cost associated with changes in value of the manipulated variablesetpoint between each of the plurality time steps.